Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = #72237167 (order 2048) · Inn(G) = C2×D8 · Out(G) = #525 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^7|c^2=d,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12278, #12280, #12284, … |
| maximal normal subgroup of | #12278, #12280, #12284, … |
| maximal quotient of | #12866, #12919, #12991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 964);